Planar electromagnetic (EM) simulation technology enables, in generalised terms, the modelling of three-dimensional electromagnetic fields excited by currents flowing in planar conductors embedded in a multilayered dielectric medium. In this technology, planar conductors that are actually three-dimensional (i.e. have a thickness that is not negligible in terms of circuit operation relative to their width) are approximated as zero thickness sheet conductors. These conductors are treated as being printed on or embedded in a stratified dielectric layer stack (also called the substrate), and as only supporting planar (two-dimensional) surface currents Js(r) that flow in the metal sheets on either horizontal or vertical planes. FIG. 1 shows a generalised planar metallization pattern (e.g. a printed circuit track or microstrip) embedded in a generalised multilayered dielectric medium (e.g. part of a multilayered printed circuit board).
In real circuits, conductors always have a finite thickness. However, for many practical RF and microwave planar structures found in monolithic microwave integrated circuits (MICs), RF boards, RF modules and planar antenna applications the metal thickness is typically much smaller than the width of the metal traces. In these cases the zero thickness model used in planar EM technology can be applied and yields valid simulation results. Moreover, in this approach the finite thickness of the conductors is not totally neglected but is taken into account through the concept of surface impedance, as described subsequently.
When considering the current distribution in the cross section of a 3D planar conductor with width w, thickness t and conductivity σ (see FIG. 2), and hence the conductor loss as a function of frequency, three frequency ranges can be distinguished (assuming that w>t), each exhibiting its own loss characteristic. At low frequencies, with skin depth large compared to width w and thickness t, the current is uniformly distributed through the cross section of the conductor and the loss is determined by the DC resistance RDC∝1/σwt. At higher frequencies the edge effect starts to manifest itself, that is, internal inductive effects tend to modify the current distribution into an exponential regime, with an increased current density at the outer edges and a decreased current density at the centre of the conductor. This so-called edge singularity effect of the current distribution increases the loss in the conductor and starts to play a role for frequencies where the width of the conductor becomes larger than twice the skin depth (w>2δs). The skin depth is defined as the depth inside the conductor at which the current density has reduced by a factor of e.
At even higher frequencies, when the conductor becomes thick compared to the skin depth (t>δs), loss increases even more because the current is increasingly confined to the surface of the conductor. This is the so-called skin-effect region in which the classic dependence of loss on the square-root of frequency occurs. For a conductor in proximity to the groundplane the current distribution is more confined to the bottom layer of the conductor, hence the skin effect is dominantly single-sided. For isolated conductors the skin effect is double-sided, i.e. the current is equally distributed and confined to both the bottom and top layers of the conductor.
In planar electromagnetic simulation the skin effect is taken into account by applying the surface impedance concept, while the edge effect is taken into account by applying the concept of an edge mesh.
The surface impedance concept is used to include resistive losses and the frequency-dependent skin effect of currents in the planar conductor modelling process. This concept is based on the decomposition of the field problem into an internal and an external field problem. The internal field problem addresses the field problem inside the conductor and yields the surface impedance relationun×E(r)=Zs(un×Js(r))  (1)between the tangential electric field at each point of the metal surface and the equivalent surface current at the same point. In the external field problem, the thick conductor is replaced by a sheet (zero-thickness) conductor supporting the single surface current Js(r) (FIG. 3) related to the tangential electric field according to expression (1).
In expression (1) E(r) is the total electric field, un is the unit vector normal to the surface of the conductor, and Zs is the frequency-dependent surface impedance of the conductor. The replacing sheet conductor is assumed to be infinitely thin and the actual thickness t and conductivity σ of the thick conductor is accounted for by the surface impedance Zs.
The Internal Field Problem
In planar EM technology two models for the surface impedance are most often used. These models are derived from the one-dimensional internal field problem associated with a finite thickness conductor plate with infinite lateral dimensions—see “Variations of microstrip losses with thickness of strip”, R. Horton, B. Easter & A. Gopinath, Electronics Letters, vol. 7, no 17, pp. 490-491, August 1971. Owing to the one-dimensional nature of the field problem, Maxwell's equation can be solved analytically yielding a closed form expression for the surface impedance. Both models yield the same and correct DC resistance of the 3D conductor but differ in the high frequency limit. The first model describes a single-sided skin effect in which all the high frequency current is confined to a single skin depth layer. In practice this situation occurs for microstrip type structures close to a groundplane such that the proximity effect of the return current in the groundplane plays an important role in the current distribution. The second model describes the double-sided skin effect in which the high frequency current is distributed over two skin depth layers, one at the top and one at the bottom of the conductor. This is the case for stripline types of structures and for microstrip types of structures where the groundplane is far enough away, such that it has no significant effect on the current distribution inside the conductor.
Single-Sided Skin Effect Model
The model for the surface impedance that describes the single-sided skin effect is given by the formula:
                                                                                                              Z                    s                                    ⁡                                      (                                          t                      ,                      σ                      ,                      ω                                        )                                                  =                                ⁢                                                      Z                    c                                    ⁢                                      coth                    ⁡                                          (                                                                        jk                          c                                                ⁢                        t                                            )                                                        ⁢                                                                          ⁢                  with                                            ⁢                                                                                                                                      Z                c                            =                            ⁢                                                jωμ                                      σ                    +                    jωɛ                                                                                                                                          jk                c                            =                            ⁢                                                jωμ                  ⁡                                      (                                          σ                      +                      jωɛ                                        )                                                                                                          (        2        )            
For low frequencies ω→0 and by applying the limit coth(z)→1/z+z/3, formula (2) is easily simplified to (3a), yielding the known DC resistance for a uniform current distribution in the entire cross section of a thick conductor with conductivity σ and thickness t:
                                          Z            s                    ⁡                      (                          t              ,              σ              ,              ω                        )                          ->                                            1                              σ                ⁢                                                                  ⁢                t                                      +                          jω              ⁢                                                μ                  ⁢                                                                          ⁢                  t                                3                            ⁢                                                          ⁢              for              ⁢                                                          ⁢              ω                                ->          0                                    (                  3          ⁢          a                )            
Note that the surface impedance (3a) also models the increased internal inductance of the single layer surface current. For a good conductor with σ>>ωe and for high frequencies where the skin depth δs is much smaller than the conductor thickness t, the limit coth(z)→1 can be applied and formula (2) is reduced to (3b), describing the known skin effect behaviour of the current distribution causing the resistance to increase and the inductance to decrease with the square root of the frequency:
                                                        Z              s                        ⁡                          (                              t                ,                σ                ,                ω                            )                                ->                                    1                              σδ                s                                      ⁢                          (                              1                +                j                            )                        ⁢                                                  ⁢            with            ⁢                                                  ⁢                          δ              s                                      =                              2            ωμσ                                              (                  3          ⁢          b                )            The conductor loss modelled by the surface impedance formula (2) is equivalent to the loss for a uniform current distribution over the entire cross section (thickness t) at low frequencies and for a concentrated current distribution over a thickness δs at high frequencies. This equivalent surface impedance yields the exact solution at low frequencies. However, at high frequencies the model only provides good results when applied for good conductors, much wider than their thickness (typically w/t>5) and in proximity to the groundplane, so that the HF currents are flowing on one side of the conductor.
Double-Sided Skin Effect Model
The model for the surface impedance that describes the double-sided skin effect is given by the formula:
                                                                                                              Z                    s                                    ⁡                                      (                                          t                      ,                      σ                      ,                      ω                                        )                                                  =                                ⁢                                                      1                    2                                    ⁢                                      Z                    c                                    ⁢                                      coth                    ⁡                                          (                                                                        jk                          c                                                ⁢                                                  1                          2                                                                    )                                                        ⁢                                                                          ⁢                  with                                            ⁢                                                                                                                                      Z                c                            =                            ⁢                                                jωμ                                      σ                    +                    jωɛ                                                                                                                                          jk                c                            =                            ⁢                                                jωμ                  ⁡                                      (                                          σ                      +                      jωɛ                                        )                                                                                                          (        4        )            In the low frequency limit for ω→0 the known DC resistance for a uniform current distribution is recovered:
                                          Z            s                    ⁡                      (                          t              ,              σ              ,              ω                        )                          ->                                            1                              σ                ⁢                                                                  ⁢                t                                      +                          jω              ⁢                                                μ                  ⁢                                                                          ⁢                  t                                12                            ⁢                                                          ⁢              for              ⁢                                                          ⁢              ω                                ->          0                                    (        5        )            In the high frequency limit where the skin depth δs is much smaller than the conductor thickness t, the limit coth(z)→1 can be applied and formula (4) is reduced to the following, describing the double-sided skin effect behaviour of the current distribution:
                                                        Z              s                        ⁡                          (                              t                ,                σ                ,                ω                            )                                ->                                    1                              2                ⁢                                  σδ                  s                                                      ⁢                          (                              1                +                j                            )                        ⁢                                                  ⁢            with            ⁢                                                  ⁢                          δ              s                                      =                              2            ωμσ                                              (        6        )            
The conductor loss modelled by the surface impedance formula (4) is equivalent to the loss of a uniform current distribution over the entire cross section (thickness t) at low frequencies, and of a double-layered current distribution over a thickness δs at high frequencies. This equivalent surface impedance yields the exact solution at low frequencies. However, at high frequencies the model only provides good results when applied for isolated good conductors, much wider than their thickness, so that the HF currents are evenly distributed and flowing both at the top and bottom sides of the conductor.
The External Field Problem
A basic integral equation in the unknown surface currents can be obtained by applying the surface impedance relation (1) above at the boundaries of the sheet conductors. By means of Green's theorem, the surface current contribution to the electric field is described by a surface integral representation in which the electric Green's dyadic of the substrate layer stack acts as the integral kernel. In the mixed potential integral equation (MPIE) formulation described in “Mixed Potential Integral Equation Technique for Hybrid Microstrip-Slotline Multilayered Circuits using a Mixed Rectangular-Triangular Mesh” by J. Sercu, N. Fache, F. Libbrecht & P. Lagasse, IEEE Transactions on Microwave Theory and Techniques, pp 1162-1172, May 1995, the electric field is decomposed into a contribution from the vector potential A(r) and a contribution from the scalar potential V(r):
                                                                        E                ⁡                                  [                                      J                    s                                    ]                                            =                                                                    -                    jω                                    ⁢                                                                          ⁢                                      A                    ⁡                                          [                                              J                        s                                            ]                                                                      -                                                      ∇                                          V                      ⁡                                              [                                                  q                          s                                                ]                                                                              ⁢                                                                          ⁢                  with                                                                                                                        A                ⁡                                  [                                      J                    s                                    ]                                            =                                                                                          ∫                      ∫                                        ⁢                                                                                                  S                                ⁢                                  ⅆ                                      S                    ′                                                  ⁢                                                                                                                              G                          _                                                _                                            A                                        ⁡                                          (                                              r                        ,                                                  r                          ′                                                                    )                                                        ·                                                            J                      s                                        ⁡                                          (                                              r                        ′                                            )                                                                                                                                                              V                ⁡                                  [                                      q                    s                                    ]                                            =                                                                    ∫                    ∫                                    S                                ⁢                                                                  ⁢                                  ⅆ                                      S                    ′                                                  ⁢                                                      G                    V                                    ⁡                                      (                                          r                      ,                                              r                        ′                                                              )                                                  ⁢                                                      q                    s                                    ⁡                                      (                                          r                      ′                                        )                                                                                                                          =                                                -                                      1                    jω                                                  ⁢                                                      ∫                    ∫                                    S                                ⁢                                                                  ⁢                                  ⅆ                                      S                    ′                                                  ⁢                                                      G                    V                                    ⁡                                      (                                          r                      ,                                              r                        ′                                                              )                                                  ⁢                                                      ∇                    s                    ′                                    ⁢                                      ·                                                                  J                        s                                            ⁡                                              (                                                  r                          ′                                                )                                                                                                                                                    (        7        )            In expression (7) GA is the dyadic magnetic Green's function and GV is the scalar electric Green's function of the multilayered medium. The scalar potential originates from the dynamic surface charge distribution qs derived from the surface current through the current continuity relation and is related to the vector potential through the Lorentz gauge.
The electric field in expression (1) is the total electric field. By splitting the total electric field into an incoming field Ein from the applied sources and a scattered field excited by the unknown surface currents, and by applying the mixed-potential decomposition (7) for the scattered field, the known mixed potential integral equation (MPIE) in the unknown surface currents is obtained:Etin(r)=jωAt[Js(r)]+∇tV[qs(r)]+ZsJs(r)  (8)Here the subscript “t” for the vectors denotes the vector component tangential to the surface S of the planar conductors.
The method of moments (MoM) solution process is used to discretise and solve the mixed potential integral equation (8). The MoM process is a numerical discretisation technique which builds a discrete approximation for the unknown surface current Js(r). This is accomplished by superimposing a notional mesh over the sheet conductor surface using rectangular and triangular cells, as shown schematically in FIG. 4. A finite number N of sub-sectional basis functions B1(r), . . . , BN(r) are defined over the mesh. They construct the basis of the discrete space within which the unknown surface current is approximated:
                                          J            s                    ⁡                      (            r            )                          =                              ∑                          j              =              1                        N                    ⁢                                    I              j                        ⁢                                          B                j                            ⁡                              (                r                )                                                                        (        9        )            The standard basis functions used in planar EM simulators are the sub-sectional “rooftop” functions defined over the rectangular and triangular cells. Each rooftop is associated with one edge of the mesh and represents a current with constant density flowing through that edge, as shown in FIG. 4. The unknown amplitudes Ij, j=1, . . . , N of the basis functions determine the currents flowing through all the edges of the mesh.
The integral equation (8) is discretised by inserting the rooftop expansion (9) of the surface currents and by applying the Galerkin testing procedure. That is, by testing the integral equation using test functions identical to the basis functions, the continuous integral equation is transformed into a discrete matrix equation:
                                          ∑                          j              =              1                        N                    ⁢                                    Z                              i                ,                j                                      ⁢                          I              j                                      =                                            V              i              in                        ⁢                                                  ⁢                                          or                ⁢                                                                  [                Z                ]                            ·                              [                I                ]                                              =                      [                          V              in                        ]                                              (        10        )            withZi,j=<Bi,jωAt[Bj]+∇tV[Bj]+ZsBj>Viin=<Bi,Esin>  (11)Here <·,·> represents the Galerkin test operator. The left-hand-side matrix [Z] is called the interaction matrix as each element in this matrix describes the electromagnetic interaction between two rooftop basis functions. The dimension N of [Z] is equal to the number of basis functions. The right-hand-side vector [Vin] represents the discretised contribution of the incoming fields generated by the sources applied at the ports of the sheet conductor. It is convenient to split the interaction elements into three parts associated with the magnetic vector potential, the electric scalar potential and the surface impedance. Thus:Zi,j=Zi,jL+Zi,jC+Zi,jR  (12a)withZi,jL=<Bi,jωAs[Bj]>Zi,jC=<Bi,∇sV[Bj]>Zi,jR=<Bi,ZsBj>  (12b)
The use of the mixed potential formulation in combination with the surface impedance concept enables decomposition of each interaction element Zi,j into an inductive term, a capacitive term and a resistive term. The inductive and capacitive interaction terms are each defined by a quadruple integral, in which the Green's functions act as integral kernels (12a); the resistive term is defined by the overlap integral of two basis functions:
                                          Z                          i              ,              j                        L                    =                      jω            ⁢                                          ∫                ∫                            S                        ⁢                                                  ⁢                                          ⅆ                                                      SB                    i                                    ⁡                                      (                    r                    )                                                              ·                                                ∫                  ∫                                S                                      ⁢                                                  ⁢                          ⅆ                              S                ′                                      ⁢                                                                                G                    _                                    _                                ⁢                                                                              A                        ⁢                                          (                                  r                  ,                                      r                    ′                                                  )                            ·                                                B                  j                                ⁡                                  (                                      r                    ′                                    )                                                                    ⁢                                  ⁢                              Z                          i              ,              j                        C                    =                                    1              jω                        ⁢                                          ∫                ∫                            S                        ⁢                                                  ⁢                          ⅆ              S                        ⁢                                          ∇                s                            ⁢                              ·                                                      B                    i                                    ⁡                                      (                    r                    )                                                                        ⁢                                          ∫                ∫                            S                        ⁢                                                  ⁢                          ⅆ                              S                ′                                      ⁢                                          G                V                            ⁡                              (                                  r                  ,                                      r                    ′                                                  )                                      ⁢                                          ∇                s                ′                            ⁢                              ·                                                      B                    j                                    ⁡                                      (                                          r                      ′                                        )                                                                                      ⁢                                  ⁢                              Z                          i              ,              j                        R                    =                                    Z              s                        ⁢                                          ∫                ∫                            S                        ⁢                                                  ⁢                                          ⅆ                                                      SB                    i                                    ⁡                                      (                    r                    )                                                              ·                                                B                  j                                ⁡                                  (                  r                  )                                                                                        (        13        )            
Implementation of the method of moments (MoM) solution process in a practical simulation consists of two major steps: loading the matrix and solving the matrix. The loading step comprises the computation of all the electromagnetic interactions between each pair of basis functions according to expressions (12a) and (12b). This involves the computation of the quadruple integrals as defined in expression (13). The computed interaction elements are stored in the interaction matrix. It is important to note that the interaction matrix as defined in the rooftop basis is a dense matrix, i.e. each rooftop function interacts with each other rooftop function. The matrix loading process is essentially a process of order N2, i.e. the computation time goes up with the square of the number of unknowns.
In the solving step the matrix equation (10) is solved for the unknown current expansion coefficients. The solution yields the amplitudes Ij, j=1, . . . , N of the rooftop basis functions which span the surface current flowing in the zero-thickness sheet conductors. Once the currents are known, the field problem is solved because all physical quantities can be expressed in terms of the currents.
Modelling of Thick Conductors
The representation of metal conductors in current planar EM technology involves certain specific approaches:                Present planar EM technology uses the concept of infinitely thin conductors. In reality this means that the thickness of the conductor should be substantially smaller than its width.        The effect of the finite thickness is accounted for by introducing the surface impedance concept (Zs).The above treatment of metal conductors turns out to be appropriate for classical microwave circuits (e.g. filters, antennas) and in many cases also for RF and digital boards.        
However, prediction of the circuit behaviour based on Maxwell's equations (and hence taking into account all effects such as capacitive and inductive crosstalk, reflection, ringing, . . . ) is no longer the sole interest of microwave and digital circuit and board designers. New and more advanced consumer devices such as GPS systems, third-generation mobile phones and game consoles, increasing demands for processing power and communication bandwidth, exchange and transport of large amounts of data (e.g. on the internet, on LAN's, WLAN's etc.), and ever more powerful computers all rely on advanced semiconductor integrated circuit (IC) technologies. The International Roadmap for Semiconductors (ITRS) predicts that the smallest IC features will shrink from 150 nm in 2002 to 50 nm by 2012, while the clock speed will increase from 1.5 GHz to 10 GHz. At the same time the importance of the interconnect (metal conductor) part of the chip will grow. In a typical advanced microprocessor, for example, total interconnection length will increase from about 2 km to about 24 km by 2012. This interconnection length also includes on-chip passive components such as capacitors and inductors.
It is quite clear that the infinitely-thin conductor approximation is not justified for on-chip interconnections. The width and thickness of the conductors is typically of the same magnitude e.g. a 1 μm by 1 μm cross-section, and the spacing between conductors is of that same order. The on-chip conductor geometry will have an impact on how the capacitive and inductive coupling between conductors should be handled. Particular attention will also have to be devoted to the resistance of the conductor. For digital signals, frequencies from DC to about 50 GHz (i.e. ten harmonics when a maximum clock rate of 10 GHz is considered) will have to be taken into account. The current distribution inside the conductor will evolve from the quasi-static case (DC resistance) to the high-frequency case (presence of skin effect and losses dominated by the surface impedance).
All these considerations lead to the conclusion that the infinitely-thin conductor approximation is no longer valid and improved modelling of the conductors is required.
Several proposals exist for incorporating homogeneous 3D conductors into an EM simulator. The most straightforward solution is to use a rigorous 3D field simulator. Such simulators are capable of handling general 3D conductor objects. However, when applied to 3D planar conductors embedded in a multilayered medium they become numerically very intensive (the discretisation size inside the conductor must typically be smaller than the skin depth) and not practical for real-life structures.
There are approaches based on the one-layer sheet conductor approximation (shown in FIG. 5(b)) and the use of either the single-sided or the double-sided model for the scalar surface impedance. A disadvantage of this approach is that these models are only valid for good conductors which are much wider than their thickness (typically w/t>5). At high frequencies both models neglect the currents on the side walls and assume a predefined distribution of the current on the top and bottom layers of the conductor. These assumptions usually lead to an overestimation of the high frequency loss. Also, with a single-layer surface current, the thickness dependency of the external inductance is neglected, so the inductance is also overestimated.
An improved high-frequency model for a thick conductor is obtained by using the two-layer sheet conductor model, shown in FIG. 6(b). In this two-layer model the volume of the conductor is divided into two equal sheet conductor layers, one at the top surface and the other at the bottom surface of the conductor. The frequency-dependent skin effect in each sheet conductor is included by using the single-sided surface impedance model (expression (2)) with a thickness equal to one half of the total conductor thickness. In this model the distribution of the current flow between the top and the bottom layer is not predefined and will follow from the solution of the EM equations, yielding an improved model for the high frequency losses. With the two-layer surface current the thickness dependency of the external inductance is included. The surface impedances used for the top and bottom layer are however uncoupled and do not account for the mutual internal inductance. Hence the two-layer surface impedance model tends to overestimate the global inductance. More details of the two-layer sheet conductor model are given in “Microstrip conductor loss models for electromagnetic analysis”, J. C. Rautio & V. Demir, IEEE Transactions on Microwave Theory and Techniques, vol. 51, no. 3, pp. 915-921, March 2003.
Other approaches are capable of handling the problem in an exact way. As already outlined above, planar EM solvers approximate the problem by introducing a scalar surface impedance that becomes inaccurate for thick conductors and is not suited for dielectrics. To handle the problem in an exact way, a typical approach described in the literature is first to solve a boundary integral equation for the fields inside the conductor, yielding a relationship between the tangential electric and magnetic fields at the boundary surface between the conductor and its surrounding layer. Next the process is repeated for the fields outside the conductor (i.e. in the layered background medium). Again a relationship is obtained between the tangential fields. Finally, by demanding the tangential field components be continuous across the boundary surface, a set of integral equations is obtained for these tangential fields. A well-known set of integral equations that are particularly suited to handle this type of problem are those known as the Poggio and Miller integral equations, described in “Integral Equation Solutions of Three Dimensional Scattering Problems” by A. J. Poggio and E. K. Miller, Computer techniques for electromagnetics, R. Mittra, Pergamon Press, Oxford, 1973. The problem can also be reformulated in terms of equivalent electric and magnetic currents residing on the boundary surface—see for example the discussion on field equivalence in “Field theory of guided waves”, R. E. Collin, IEEE Press Series on Electromagnetic Waves, 1990. This approach has two major disadvantages when used in conjunction with a planar EM solver:                The need to introduce both equivalent electric and magnetic surface currents doubles the number of unknowns that have to be handled by the solver as compared to the case where only electric surface currents are needed.        Present planar solver technology is inherently based on the concept that a conductor surface is modelled by electric surface currents alone. The Green's functions used in the solver calculate the electric field associated with these currents, at the same time crucially ensuring that the tangential magnetic field satisfies the correct jump condition at the current layer (see FIG. 7): un×H1−un×H2=Js. The tangential electric field remains continuous, i.e. un×E1=un×E2. The magnetic field itself is never calculated. In order to apply the approach explained above, it would become necessary to know these magnetic fields as well (and hence the curl of the magnetic vector potential).        